The HS2 system arises in the short-wave (or high-frequency) limits,
obtained via the space-time scaling
\( (x,t)\mapsto (\varepsilon x, \varepsilon t) \) 
and letting \(\varepsilon \)  tend to zero in the resulting equation,
of the two-component integrable Camassa-Holm system [1]}, [2]}, [3]}, [4]}.
This system, reading as
(REF ) with \(m\)  replaced by \((1-\partial _{xx}^2)u\) ,
was derived from the Green-Naghdi equations, which are approximations to the governing
equations for water waves. It has recently been the object of intensive study,
see, e.g., [5]}, [1]}, [7]}, [8]}, [9]}.
The HS2 system (REF ) is integrable, it has Lax pair [1]} and
a bi-Hamiltonian structure [1]}, [3]}.
It is also a particular
case of the Gurevich-Zybin system describing the dynamics in a model of nondissipative
dark matter [13]}, [14]}, [15]}.
The mathematical properties of Eq.(REF ) have been studied further in many works,
see, e.g., [1]}, [17]}, [13]}, [19]}, [20]}, [4]}, [15]}.
